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Let's go through the problem step by step. The image outlines the following tasks:

We are given a sample space \( S = \{E_1, E_2, E_3, E_4, E_5, E_6, E_7\} \) with associated probabilities for each event:

\[
P(E_1) = 0.10, \quad P(E_2) = 0.20, \quad P(E_3) = 0.15, \quad P(E_4) = 0.25, \quad P(E_5) = 0.10, \quad P(E_6) = 0.05, \quad P(E_7) = 0.15
\]

You are also given three events:
- \( A = \{E_1, E_4, E_5\} \)
- \( B = \{E_3, E_6, E_7\} \)
- \( C = \{E_4, E_5, E_7\} \)

### Task (a): Find \( P(A) \), \( P(B) \), and \( P(C) \).

#### \( P(A) \)
\[
P(A) = P(E_1) + P(E_4) + P(E_5) = 0.10 + 0.25 + 0.10 = 0.45
\]

#### \( P(B) \)
\[
P(B) = P(E_3) + P(E_6) + P(E_7) = 0.15 + 0.05 + 0.15 = 0.35
\]

#### \( P(C) \)
\[
P(C) = P(E_4) + P(E_5) + P(E_7) = 0.25 + 0.10 + 0.15 = 0.50
\]

### Task (b): What is \( A \cup B \)?

The union of \( A \) and \( B \) is the set of all elements that are in either \( A \) or \( B \):

\[
A \cup B = \{E_1, E_3, E_4, E_5, E_6, E_7\}
\]

Hence, \( P(A \cup B) \) is:

\[
P(A \cup B) = P(E_1) + P(E_3) + P(E_4) + P(E_5) + P(E_6) + P(E_7) = 0.10 + 0.15 + 0.25 + 0.10 + 0.05 + 0.15 = 0.80
\]

### Task (c): What is \( A \cap B \)?

The intersection of \( A \) and \( B \) is the set of elements that are in both \( A \) and \( B \):

\[
A \cap B = \varnothing
\]

Since the intersection is the empty set, \( P(A \cap B) = 0 \).

### Task (d): Are events \( A \) and \( C \) mutually exclusive?

Two events are mutually exclusive if they have no elements in common, i.e., their intersection is the empty set. Let's find \( A \cap C \):

\[
A \cap C = \{E_4, E_5\}
\]

Since \( A \cap C \neq \varnothing \), the events \( A \) and \( C \) are **not** mutually exclusive.

---

If you would like further details or clarification on any of the steps, feel free to ask! Here are 5 follow-up questions related to this problem:

1. What is the probability of the complement of event \( A \)?
2. How do you calculate \( P(A \cup C) \)?
3. Are events \( B \) and \( C \) mutually exclusive?
4. Can we apply the addition rule for \( P(A \cup B) \)?
5. How would you approach finding \( P(A \cap C) \) if more events were added to \( A \) or \( C \)?

**Tip:** Remember, when calculating the union of two events, make sure not to double-count any elements that belong to both events!

Suppose that we have a sample space S = {E1, E2, E3, E4, E5, E6, E7}, where probabilities for each event and specific sets A, B, and C are provided. Tasks involve calculating probabilities of individual events, unions, and intersections, as well as determining whether events are mutually exclusive.

Explore how to calculate probabilities of events, unions, and intersections in a sample space S = {E1, E2, E3, E4, E5, E6, E7}. Learn about mutually exclusive events and the addition rule of probability. Suitable for college or advanced high school students.

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